\[ \begin{align} p(c=\mathrm{b} | x) =& \int \cdots \int d\mu_b d\mu_p d\sigma^2_b d\sigma^2_p \\ & \frac{p(x | \mu_b, \mu_p) p(c=b)}{p(x | \mu_b, \mu_p) p(c=b) + p(x | \mu_p, \mu_p) p(c=p)} \\ & p(\mu_b, \mu_p, \sigma^2_b, \sigma^2_p) \end{align} \]